Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online.The word linear comes from the Latin word linearis, which means created by lines. In mathematics, a linear map or function f(x) is a function which satisfies the following two properties... Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition. Homogeneity of degree 1: f(¿x) = ¿f(x) for all ¿. It turns out that homogeneity follows from the additivity property in all cases where ¿ is rational. (proof) In that case, provided that the function is continuous, it becomes useless to establish the condition of homogeneity as an additional axiom. In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics. The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian.